Spun off from #14175 (closed)##14230 (closed). This program, which can only really be compiled on GHC HEAD at the moment:

{-# LANGUAGE GADTs #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TypeFamilies #-}
module Bug where

class C k where
  data CD :: k -> k -> *

instance C (Maybe a) where
  data CD :: (k -> *) -> (k -> *) -> *

Gives a heinous error message:

Bug.hs:11:3: error:
    • Expected kind ‘(k -> *) -> (k -> *) -> *’,
        but ‘CD :: (k -> *) -> (k -> *) -> *’ has kind ‘Maybe a
                                                        -> Maybe a -> *’
    • In the data instance declaration for ‘CD’
      In the instance declaration for ‘C (Maybe a)’
   |
11 |   data CD :: (k -> *) -> (k -> *) -> *
   |   ^^^^^^^

• We shouldn't be expecting kind (k -> *) -> (k -> *) -> *, but rather kind Maybe a -> Maybe a -> *, due to the instance head itself.
• The phrase ‘CD :: (k -> *) -> (k -> *) -> *’ has kind ‘Maybe -> Maybe a -> *’ is similarly confusing. This doesn't point out that the real issue is the use of (k -> *).

Another program in a similar vein:

{-# LANGUAGE GADTs #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeInType #-}
module Bug where

import Data.Kind

class C a where
  data CD k (a :: k) :: k -> *

instance C (Maybe a) where
  data CD k (a :: k -> *) :: (k -> *) -> *

Bug.hs:13:3: error:
    • Expected kind ‘(k -> *) -> *’,
        but ‘CD k (a :: k -> *) :: (k -> *) -> *’ has kind ‘k -> *’
    • In the data instance declaration for ‘CD’
      In the instance declaration for ‘C (Maybe a)’
   |
13 |   data CD k (a :: k -> *) :: (k -> *) -> *
   |   ^^^^^^^^^^^^^^^^^^^^^^^

This error message is further muddled by the incorrect use of k as the first type pattern (instead of k -> *, as subsequent kind signatures would suggest).